Integrand size = 21, antiderivative size = 138 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}-\frac {5 a^3 \sec ^3(c+d x)}{3 d}-\frac {5 a^3 \sec ^4(c+d x)}{4 d}+\frac {a^3 \sec ^5(c+d x)}{5 d}+\frac {a^3 \sec ^6(c+d x)}{2 d}+\frac {a^3 \sec ^7(c+d x)}{7 d} \] Output:
-a^3*ln(cos(d*x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d-5/3*a^3*se c(d*x+c)^3/d-5/4*a^3*sec(d*x+c)^4/d+1/5*a^3*sec(d*x+c)^5/d+1/2*a^3*sec(d*x +c)^6/d+1/7*a^3*sec(d*x+c)^7/d
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=-\frac {a^3 (-3732-4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))-2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))-630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (8+35 \log (\cos (c+d x)))) \sec ^7(c+d x)}{6720 d} \] Input:
Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]
Output:
-1/6720*(a^3*(-3732 - 4522*Cos[2*(c + d*x)] + 1050*Cos[3*(c + d*x)] - 2380 *Cos[4*(c + d*x)] - 210*Cos[5*(c + d*x)] - 630*Cos[6*(c + d*x)] + 2205*Cos [3*(c + d*x)]*Log[Cos[c + d*x]] + 735*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[7*(c + d*x)]*Log[Cos[c + d*x]] + 105*Cos[c + d*x]*(8 + 35*Log[Cos [c + d*x]]))*Sec[c + d*x]^7)/d
Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan ^5(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3dx\) |
| \(\Big \downarrow \) 4367 |
| \(\displaystyle -\frac {\int a^7 (1-\cos (c+d x))^2 (\cos (c+d x)+1)^5 \sec ^8(c+d x)d\cos (c+d x)}{a^4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int (1-\cos (c+d x))^2 (\cos (c+d x)+1)^5 \sec ^8(c+d x)d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {a^3 \int \left (\sec ^8(c+d x)+3 \sec ^7(c+d x)+\sec ^6(c+d x)-5 \sec ^5(c+d x)-5 \sec ^4(c+d x)+\sec ^3(c+d x)+3 \sec ^2(c+d x)+\sec (c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \left (-\frac {1}{7} \sec ^7(c+d x)-\frac {1}{2} \sec ^6(c+d x)-\frac {1}{5} \sec ^5(c+d x)+\frac {5}{4} \sec ^4(c+d x)+\frac {5}{3} \sec ^3(c+d x)-\frac {1}{2} \sec ^2(c+d x)-3 \sec (c+d x)+\log (\cos (c+d x))\right )}{d}\) |
Input:
Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^5,x]
Output:
-((a^3*(Log[Cos[c + d*x]] - 3*Sec[c + d*x] - Sec[c + d*x]^2/2 + (5*Sec[c + d*x]^3)/3 + (5*Sec[c + d*x]^4)/4 - Sec[c + d*x]^5/5 - Sec[c + d*x]^6/2 - Sec[c + d*x]^7/7))/d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.49 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{6}}{2}+\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {5 \sec \left (d x +c \right )^{4}}{4}-\frac {5 \sec \left (d x +c \right )^{3}}{3}+\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(84\) |
| default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{6}}{2}+\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {5 \sec \left (d x +c \right )^{4}}{4}-\frac {5 \sec \left (d x +c \right )^{3}}{3}+\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(84\) |
| parts | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {2 \sec \left (d x +c \right )^{5}}{5}+\frac {\sec \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {3 a^{3} \left (\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {2 \sec \left (d x +c \right )^{3}}{3}+\sec \left (d x +c \right )\right )}{d}+\frac {a^{3} \tan \left (d x +c \right )^{6}}{2 d}\) | \(132\) |
| risch | \(i a^{3} x +\frac {2 i a^{3} c}{d}+\frac {2 a^{3} \left (315 \,{\mathrm e}^{13 i \left (d x +c \right )}+105 \,{\mathrm e}^{12 i \left (d x +c \right )}+1190 \,{\mathrm e}^{11 i \left (d x +c \right )}-525 \,{\mathrm e}^{10 i \left (d x +c \right )}+2261 \,{\mathrm e}^{9 i \left (d x +c \right )}-420 \,{\mathrm e}^{8 i \left (d x +c \right )}+3732 \,{\mathrm e}^{7 i \left (d x +c \right )}-420 \,{\mathrm e}^{6 i \left (d x +c \right )}+2261 \,{\mathrm e}^{5 i \left (d x +c \right )}-525 \,{\mathrm e}^{4 i \left (d x +c \right )}+1190 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{2 i \left (d x +c \right )}+315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(204\) |
Input:
int((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x,method=_RETURNVERBOSE)
Output:
a^3/d*(1/7*sec(d*x+c)^7+1/2*sec(d*x+c)^6+1/5*sec(d*x+c)^5-5/4*sec(d*x+c)^4 -5/3*sec(d*x+c)^3+1/2*sec(d*x+c)^2+3*sec(d*x+c)+ln(sec(d*x+c)))
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=-\frac {420 \, a^{3} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 1260 \, a^{3} \cos \left (d x + c\right )^{6} - 210 \, a^{3} \cos \left (d x + c\right )^{5} + 700 \, a^{3} \cos \left (d x + c\right )^{4} + 525 \, a^{3} \cos \left (d x + c\right )^{3} - 84 \, a^{3} \cos \left (d x + c\right )^{2} - 210 \, a^{3} \cos \left (d x + c\right ) - 60 \, a^{3}}{420 \, d \cos \left (d x + c\right )^{7}} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="fricas")
Output:
-1/420*(420*a^3*cos(d*x + c)^7*log(-cos(d*x + c)) - 1260*a^3*cos(d*x + c)^ 6 - 210*a^3*cos(d*x + c)^5 + 700*a^3*cos(d*x + c)^4 + 525*a^3*cos(d*x + c) ^3 - 84*a^3*cos(d*x + c)^2 - 210*a^3*cos(d*x + c) - 60*a^3)/(d*cos(d*x + c )^7)
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (121) = 242\).
Time = 1.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.85 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=\begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{7 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{5 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} - \frac {4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{5 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {8 a^{3} \sec ^{3}{\left (c + d x \right )}}{105 d} + \frac {a^{3} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {8 a^{3} \sec {\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right )^{3} \tan ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**5,x)
Output:
Piecewise((a**3*log(tan(c + d*x)**2 + 1)/(2*d) + a**3*tan(c + d*x)**4*sec( c + d*x)**3/(7*d) + a**3*tan(c + d*x)**4*sec(c + d*x)**2/(2*d) + 3*a**3*ta n(c + d*x)**4*sec(c + d*x)/(5*d) + a**3*tan(c + d*x)**4/(4*d) - 4*a**3*tan (c + d*x)**2*sec(c + d*x)**3/(35*d) - a**3*tan(c + d*x)**2*sec(c + d*x)**2 /(2*d) - 4*a**3*tan(c + d*x)**2*sec(c + d*x)/(5*d) - a**3*tan(c + d*x)**2/ (2*d) + 8*a**3*sec(c + d*x)**3/(105*d) + a**3*sec(c + d*x)**2/(2*d) + 8*a* *3*sec(c + d*x)/(5*d), Ne(d, 0)), (x*(a*sec(c) + a)**3*tan(c)**5, True))
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=-\frac {420 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, a^{3} \cos \left (d x + c\right )^{6} + 210 \, a^{3} \cos \left (d x + c\right )^{5} - 700 \, a^{3} \cos \left (d x + c\right )^{4} - 525 \, a^{3} \cos \left (d x + c\right )^{3} + 84 \, a^{3} \cos \left (d x + c\right )^{2} + 210 \, a^{3} \cos \left (d x + c\right ) + 60 \, a^{3}}{\cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="maxima")
Output:
-1/420*(420*a^3*log(cos(d*x + c)) - (1260*a^3*cos(d*x + c)^6 + 210*a^3*cos (d*x + c)^5 - 700*a^3*cos(d*x + c)^4 - 525*a^3*cos(d*x + c)^3 + 84*a^3*cos (d*x + c)^2 + 210*a^3*cos(d*x + c) + 60*a^3)/cos(d*x + c)^7)/d
Time = 0.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=-\frac {420 \, a^{3} \log \left ({\left | \cos \left (d x + c\right ) \right |}\right ) - \frac {1260 \, a^{3} \cos \left (d x + c\right )^{6} + 210 \, a^{3} \cos \left (d x + c\right )^{5} - 700 \, a^{3} \cos \left (d x + c\right )^{4} - 525 \, a^{3} \cos \left (d x + c\right )^{3} + 84 \, a^{3} \cos \left (d x + c\right )^{2} + 210 \, a^{3} \cos \left (d x + c\right ) + 60 \, a^{3}}{\cos \left (d x + c\right )^{7}}}{420 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm="giac")
Output:
-1/420*(420*a^3*log(abs(cos(d*x + c))) - (1260*a^3*cos(d*x + c)^6 + 210*a^ 3*cos(d*x + c)^5 - 700*a^3*cos(d*x + c)^4 - 525*a^3*cos(d*x + c)^3 + 84*a^ 3*cos(d*x + c)^2 + 210*a^3*cos(d*x + c) + 60*a^3)/cos(d*x + c)^7)/d
Time = 16.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.60 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=\frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {422\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {382\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {352\,a^3}{105}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \] Input:
int(tan(c + d*x)^5*(a + a/cos(c + d*x))^3,x)
Output:
(2*a^3*atanh(tan(c/2 + (d*x)/2)^2))/d - ((422*a^3*tan(c/2 + (d*x)/2)^4)/5 - (382*a^3*tan(c/2 + (d*x)/2)^2)/15 - (224*a^3*tan(c/2 + (d*x)/2)^6)/3 + ( 128*a^3*tan(c/2 + (d*x)/2)^8)/3 - 14*a^3*tan(c/2 + (d*x)/2)^10 + 2*a^3*tan (c/2 + (d*x)/2)^12 + (352*a^3)/105)/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/ 2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*ta n(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1) )
Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.26 \[ \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx=\frac {a^{3} \left (210 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right )+60 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{4}-48 \sec \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}+32 \sec \left (d x +c \right )^{3}+210 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{4}-210 \sec \left (d x +c \right )^{2} \tan \left (d x +c \right )^{2}+210 \sec \left (d x +c \right )^{2}+252 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{4}-336 \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}+672 \sec \left (d x +c \right )+105 \tan \left (d x +c \right )^{4}-210 \tan \left (d x +c \right )^{2}\right )}{420 d} \] Input:
int((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x)
Output:
(a**3*(210*log(tan(c + d*x)**2 + 1) + 60*sec(c + d*x)**3*tan(c + d*x)**4 - 48*sec(c + d*x)**3*tan(c + d*x)**2 + 32*sec(c + d*x)**3 + 210*sec(c + d*x )**2*tan(c + d*x)**4 - 210*sec(c + d*x)**2*tan(c + d*x)**2 + 210*sec(c + d *x)**2 + 252*sec(c + d*x)*tan(c + d*x)**4 - 336*sec(c + d*x)*tan(c + d*x)* *2 + 672*sec(c + d*x) + 105*tan(c + d*x)**4 - 210*tan(c + d*x)**2))/(420*d )